

from __future__ import annotations

from random import random







class     __lowercase :
 def __init__(				self ,				A_ = None   )   ->List[Any]:



       '''simple docstring'''





       __lowerCAmelCase   :		Optional[int]   =   value
       __lowerCAmelCase   :		Dict   =   random()
       __lowerCAmelCase   :		str   =   None
       __lowerCAmelCase   :		List[str]   =   None
 def __repr__(				self   )   ->Union[str, Any]:



       '''simple docstring'''





       from pprint import pformat

       if self.left is None and self.right is None:
             return f"""\'{self.value}: {self.prior:.5}\'"""
       else:
             return pformat(
                 {f"""{self.value}: {self.prior:.5}""": (self.left, self.right)} ,				indent=1   )





 def __str__(				self   )   ->str:



       '''simple docstring'''





       __lowerCAmelCase   :		int   =   str(self.value   ) + ''' '''
       __lowerCAmelCase   :		List[str]   =   str(self.left or ''''''   )
       __lowerCAmelCase   :		Optional[int]   =   str(self.right or ''''''   )
       return value + left + right







def 						_lowercase					(	lowercase__  ,      lowercase__		):






      if root is None:  # None tree is split into 2 Nones
            return None, None
      elif root.value is None:
            return None, None
      else:
            if value < root.value:
                  __lowerCAmelCase,				__lowerCAmelCase   :		Optional[Any]   =   split(root.left  ,      UpperCamelCase_		)
                  return left, root
            else:
                  __lowerCAmelCase,				__lowerCAmelCase   :		List[Any]   =   split(root.right  ,      UpperCamelCase_		)
                  return root, right







def 						_lowercase					(	lowercase__  ,      lowercase__		):






      if (not left) or (not right):  # If one node is None, return the other
            return left or right
      elif left.prior < right.prior:
            __lowerCAmelCase   :		int   =   merge(left.right  ,      UpperCamelCase_		)
            return left
      else:
            __lowerCAmelCase   :		Optional[int]   =   merge(UpperCamelCase_  ,      right.left		)
            return right







def 						_lowercase					(	lowercase__  ,      lowercase__		):
      __lowerCAmelCase   :		List[Any]   =   Node(UpperCamelCase_		)
      __lowerCAmelCase,				__lowerCAmelCase   :		List[str]   =   split(UpperCamelCase_  ,      UpperCamelCase_		)
      return merge(merge(UpperCamelCase_  ,      UpperCamelCase_		)  ,      UpperCamelCase_		)







def 						_lowercase					(	lowercase__  ,      lowercase__		):
      __lowerCAmelCase,				__lowerCAmelCase   :		Dict   =   split(UpperCamelCase_  ,      value - 1		)
      __lowerCAmelCase,				__lowerCAmelCase   :		int   =   split(UpperCamelCase_  ,      UpperCamelCase_		)
      return merge(UpperCamelCase_  ,      UpperCamelCase_		)







def 						_lowercase					(	lowercase__		):






      if not root:  # None
            return
      else:
            inorder(root.left		)
            print(root.value  ,      end=''','''		)
            inorder(root.right		)







def 						_lowercase					(	lowercase__  ,      lowercase__		):






      for arg in args.split():
            if arg[0] == "+":
                  __lowerCAmelCase   :		Tuple   =   insert(UpperCamelCase_  ,      int(arg[1:]		)		)

            elif arg[0] == "-":
                  __lowerCAmelCase   :		Any   =   erase(UpperCamelCase_  ,      int(arg[1:]		)		)

            else:
                  print('''Unknown command'''		)

      return root







def 						_lowercase					(	):
      __lowerCAmelCase   :		List[Any]   =   None
      print(
          '''enter numbers to create a tree, + value to add value into treap, '''
          '''- value to erase all nodes with value. \'q\' to quit. '''		)

      __lowerCAmelCase   :		Optional[Any]   =   input()
      while args != "q":
            __lowerCAmelCase   :		int   =   interact_treap(UpperCamelCase_  ,      UpperCamelCase_		)
            print(UpperCamelCase_		)
            __lowerCAmelCase   :		int   =   input()

      print('''good by!'''		)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 main()






from math import factorial

class     A__   :







    """simple docstring"""

    def __init__(			self     ,   __snake_case     ,   __snake_case       ):
         snake_case											=							real
         if isinstance(__snake_case     ,   __snake_case       ):
              snake_case											=							[1] * rank
         else:
              snake_case											=							rank
    def __repr__(			self       ):
         return (
             F'''{self.real}+'''
             F'''{"+".join(str(__snake_case       )+"E"+str(n+1       )for n,dual in enumerate(self.duals       )       )}'''
         )
    def 						a_     (			self       ):
         snake_case											=							self.duals.copy()
         while cur[-1] == 0:
              cur.pop(-1       )
         return Dual(self.real     ,   __snake_case       )
    def __add__(			self     ,   __snake_case       ):
         if not isinstance(__snake_case     ,   __snake_case       ):
              return Dual(self.real + other     ,   self.duals       )
         snake_case											=							self.duals.copy()
         snake_case											=							other.duals.copy()
         if len(__snake_case       ) > len(__snake_case       ):
              o_dual.extend([1] * (len(__snake_case       ) - len(__snake_case       ))       )
         elif len(__snake_case       ) < len(__snake_case       ):
              s_dual.extend([1] * (len(__snake_case       ) - len(__snake_case       ))       )
         snake_case											=							[]
         for i in range(len(__snake_case       )       ):
              new_duals.append(s_dual[i] + o_dual[i]       )
         return Dual(self.real + other.real     ,   __snake_case       )

    __magic_name__         =						__add__
    def __sub__(			self     ,   __snake_case       ):
         return self + other * -1
    def __mul__(			self     ,   __snake_case       ):
         if not isinstance(__snake_case     ,   __snake_case       ):
              snake_case											=							[]
              for i in self.duals:
                   new_duals.append(i * other       )
              return Dual(self.real * other     ,   __snake_case       )
         snake_case											=							[0] * (len(self.duals       ) + len(other.duals       ) + 1)
         for i, item in enumerate(self.duals       ):
              for j, jtem in enumerate(other.duals       ):
                   new_duals[i + j + 1] += item * jtem
         for k in range(len(self.duals       )       ):
              new_duals[k] += self.duals[k] * other.real
         for index in range(len(other.duals       )       ):
              new_duals[index] += other.duals[index] * self.real
         return Dual(self.real * other.real     ,   __snake_case       )

    __magic_name__         =						__mul__
    def __truediv__(			self     ,   __snake_case       ):
         if not isinstance(__snake_case     ,   __snake_case       ):
              snake_case											=							[]
              for i in self.duals:
                   new_duals.append(i / other       )
              return Dual(self.real / other     ,   __snake_case       )
         raise ValueError
    def __floordiv__(			self     ,   __snake_case       ):
         if not isinstance(__snake_case     ,   __snake_case       ):
              snake_case											=							[]
              for i in self.duals:
                   new_duals.append(i // other       )
              return Dual(self.real // other     ,   __snake_case       )
         raise ValueError




    def __pow__(			self     ,   __snake_case       ):
         if n < 0 or isinstance(__snake_case     ,   __snake_case       ):
              raise ValueError('''power must be a positive integer'''       )
         if n == 0:
              return 1
         if n == 1:
              return self
         snake_case											=							self
         for _ in range(n - 1       ):
              x *= self
         return x






def      UpperCAmelCase__		(UpperCamelCase_				,UpperCamelCase_				,UpperCamelCase_							):



     """simple docstring"""






     if not callable(UpperCamelCase_							):
          raise ValueError('''differentiate() requires a function as input for func'''							)
     if not isinstance(UpperCamelCase_				,(float, int)							):
          raise ValueError('''differentiate() requires a float as input for position'''							)
     if not isinstance(UpperCamelCase_				,UpperCamelCase_							):
          raise ValueError('''differentiate() requires an int as input for order'''							)
     snake_case											=							Dual(UpperCamelCase_				,1							)
     snake_case											=							func(UpperCamelCase_							)
     if order == 0:
          return result.real
     return result.duals[order - 1] * factorial(UpperCamelCase_							)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 def      UpperCAmelCase__		(UpperCamelCase_							):



      """simple docstring"""






      return y**2 * y**4

 print(differentiate(f, 9, 2))




